Question: A culture of bacteria starts with $50$ bacteria and increases exponentially. The relationship between $B$, the number of bacteria in the culture, and $d$, the elapsed time, in days, is modeled by the following equation. B = 50 ⋅ 10 d 2 B=50\cdot 10\^{{\frac{d}2}} In how many days will the number of bacteria in the culture reach $800{,}000$ ? Give an exact answer expressed as a base-ten logarithm. days
Answer: Thinking about the problem We want to know how many days, $d$, it will take for the number of bacteria in the culture, $B$, to reach $800{,}000$. So we need to find the value of $d$ for which $B=800{,}000$. Substituting $800{,}000$ in for $B$ in the model gives us the following equation. 800,000 = 50 ⋅ 10 d 2 800{,}000=50\cdot 10\^{{\frac{d}2}} Solving the equation We can solve the equation as shown below. 50 ⋅ 10 d 2 10 d 2 d 2 d = 800,000 = 16,000 = log ( 16,000 ) = 2 log ( 16,000 ) \begin{aligned}50\cdot 10\^{{\frac{d}2}}&=800{,}000\\\\ 10\^{{\frac{d}2}}&=16{,}000\\\\ \dfrac d2&=\log\left(16{,}000\right)\\\\ d&=2{\,\log\left(16{,}000\right)}\\\\ \end{aligned} It will take $2{\,\log\left(16{,}000\right)}$ days for the number of bacteria in the culture to reach $800{,}000$. The expression above represents an exact solution to the equation. We can use a calculator to approximate the value of the expression, but this will be a rounded inexact answer. The answer The answer is $2\,{\log\left(16000\right)}$ days.